System for performing mulitplication and division in GF(2 2m)

ABSTRACT

A system determines the multiplicative inverse of A∈GF(2 2M ) by representing A using a selected basis in which basis elements are squares of one another, and performing various operations that involve raising A to powers of 2 as cyclic rotations of A. The system also performs multiplication operations over GF(2 2M ) or subfields thereof by calculating the coefficients of the product of two elements A and B that are represented using the selected basis as combinations of the coefficients of cyclically rotated versions of A and B. The system further utilizes a relatively small look-up table that contains the multiplicative inverses of selected elements of a subfield of GF(2 2M ). The system may then cyclically rotate the multiplicative inverse values read from the table to produce the multiplicative inverses of the remaining elements of the subfield. Thereafter, as applicable, the system further manipulates the multiplicative inverse of the subfield element, to produce the multiplicative inverse of the desired element of GF(2 2M ). Using the selected basis, elements of GF(2 2M ) that are elements of the subfields have m lowest-order coefficients that are duplicates of the m highest order coefficients. Each element in the look-up table can thus be represented using only m bits, and the table can be entered with m bits.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] This invention relates to data error correction systems, and in particular to Galois Field division operations performed by the systems.

[0003] 2.Background Information

[0004] Data stored on magnetic media, such as a magnetic tape and disks, are typically stored in encoded form, so that errors in the stored data can possibly be corrected. The errors may occur, for example, because of inter-symbol interference, a defect in the tape or disk, or noise. As the density of the data stored on the tape or disk increases, more errors are likely, and the system is required to correct greater numbers of errors. The speed with which the system corrects the errors is important to the overall speed with which the system processes the data.

[0005] Before a string of data symbols is recorded, it is mathematically encoded to form ECC symbols. The ECC symbols are then appended to the data string to form code words—data symbols plus ECC symbols—and the code words are then stored. When the stored data is to be accessed, the code words containing the data symbols are retrieved and mathematically decoded. During decoding any errors in the data are detected and, if possible, corrected through manipulation of the ECC symbols [For a detailed description of decoding see Peterson and Weldon, Error Correction Codes, 2d Edition, MIT Press, 1972].

[0006] Stored digital data can contain multiple errors. One of the most effective types of ECC used for the correction of multiple errors is a Reed-Solomon code [For a detailed description of Reed-Solomon codes, see Peterson and Weldon, Error Correction Codes]. With Reed-Solomon codes, the encoding and decoding operations are performed over a Galois Field, using Galois Field addition, multiplication and division operations.

[0007] Galois Field division is generally a time consuming operation that significantly lengthens the ECC encoding and decoding processes. The time it takes to perform error correction operations adversely affects the rate at which data can be retrieved from a storage device and supplied to a user or to a computer application that requires the data.

[0008] One way to perform a Galois Field division operation, B/A, where A and B are elements of the Galois Field GF(2^(Q)), is to determine the multiplicative inverse of A and multiply B by the inverse. Finding the inverse of a Galois Field element is not particularly straightforward. One solution to determining the inverse is to test each element of the field, by multiplying the element by A. This is very time consuming, particularly with the larger Galois Fields that are used to protect the higher-density data.

[0009] Another solution is to use a look-up table that contains the multiplicative inverses. If a Galois Field GF(2^(2M)) is used, a (2^(2M)−1)-element look-up table is required. With many systems, it is undesirable to have such large look-up tables, which require both large amounts of storage space and relatively complex addressing circuitry.

[0010] A better solution is described in U.S. Pat. No. 4,975,876, which has a common inventor and is incorporated herein by reference. The system described in the '876 patent determines the multiplicative inverse of an element A of GF(2^(2M)) by first computing a conversion factor, D=A² ^(M) , and then multiplying A by D to produce an associated element C=A² ^(M) ⁺¹. The element C is also an element of a smaller Galois Field, GF(2^(M)), which is a subfield of GF(2^(2M)). The system then determines the multiplicative inverse of C by entering a (2^(M)−1)-element look-up table. The system next converts the inverse of C, C⁻¹=A⁻⁽² ^(M) ⁺¹), to the multiplicative inverse of A by multiplying C⁻¹ by the conversion factor D, to produce A⁻⁽² ^(M) ⁺¹)*A² ^(M) =A⁻¹, where “*” represents Galois Field multiplication.

[0011] The system described in the '876 patent works well and determines the multiplicative inverse of A relatively quickly, using a (2^(M)−1)-entry look-up table rather than the larger (2^(2M)−1)-entry table. The system, however, requires at least one relatively complex fall Galois Field multiplier to multiply together two 2M-bit symbols. A single multiplier may be used multiple times to multiply together A and D, and then C⁻¹ and D, or two of the multipliers may be used to perform the two multiplications. The multiplication operations, whether performed by one or two full Galois Field multipliers, involve manipulation of two 2M-bit symbols and are thus both relatively time consuming and complex.

[0012] We have devised an improved system that produces the multiplicative inverses of the elements of GF(2^(2M)) in fewer clock cycles and/or using less complex multipliers, and/or using smaller tables. We discuss the improved system below.

SUMMARY OF THE INVENTION

[0013] The invention is a system that determines the multiplicative inverse of A∈GF(2^(2M)) by representing A using a selected basis and performing various operations that involve raising A to powers of 2 as cyclic rotations of A. The system also performs multiplication operations over GF(2^(2M)) or subfields thereof by calculating the coefficients of the product of two elements A and B that are represented using the selected basis as combinations of the coefficients of cyclically rotated versions of A and B.

[0014] The system utilizes a relatively small look-up table that contains the multiplicative inverses of selected elements of a subfield of GF(2^(2M)). The system may then cyclically rotate the multiplicative inverse values read from the table to produce the multiplicative inverses of the remaining elements of the subfield. Thereafter, as applicable, the system further manipulates the multiplicative inverse of the subfield element, to produce the multiplicative inverse of the desired element of GF(2^(2M)).

[0015] More specifically, the system represents the elements of GF(2^(2M)) using a selected basis in which each basis element is the square of the preceding basis element. Raising a field element to a power 2^(k) is then a cyclic rotation of the element by k bits. Further, multiplication of A*B, where B is also represented using the selected basis, is performed essentially by combining cyclically rotated versions of A and B, as discussed in more detail below.

[0016] The system thus raises A to the power 2^(M) by cyclic rotation of A, or the equivalent thereof, to produce the conversion factor A² ^(M) and multiplies the factor by A in the manner discussed above to produce A² ^(M+1) , which is an element of the subfield GF(2^(M)). The multiplicative inverse of this element may then be determined by entering a look-up table that contains the multiplicative inverses of elements of GF(2^(M)). Using the selected basis, elements of GF(2^(2M)) that are elements of the subfield GF(2^(M)) have m lowest-order coefficients that are duplicates of the m highest order coefficients. Each element in the look-up table can thus be represented using only m bits, and the table can be entered with m bits. Accordingly, the table and the associated circuitry are less complex than those required when the conventional basis is used to represent the elements.

[0017] After retrieving the multiplicative inverse from the table, the system then multiplies it by the conversion factor to produce A⁻¹. As discussed, the multiplication operations are readily performed using cyclically rotated versions of the elements.

[0018] The system may further manipulate A² ^(M) ⁺¹ by raising the element to selected powers of 2, to produce elements that are also elements of smaller subfields of GF(2^(2M)), e.g., $G\quad {F\left( 2^{\frac{M}{2}} \right)}$

[0019] and so forth. The system may then use correspondingly smaller look-up tables.

[0020] The size of the tables can be even further reduced by one-half or more by including therein only the multiplicative inverses of the elements of a given subfield that have one or more coefficients set to particular values. As an example, the table may include the multiplicative inverses of the elements that have the highest-order coefficient set to a 1. To enter the table, the system raises the remaining elements of the subfield to powers of 2, that is, cyclically rotates the elements to the left, to produce elements that have the particular coefficients set to the desired pattern. Thereafter, the system rotates the multiplicative inverse read from the table by the same number of bits to the right, or raises the result to powers of $\frac{1}{2},$

[0021] to produce the multiplicative inverse of the desired element of the subfield. As appropriate, the system further manipulates the result to produce the multiplicative inverse of the associated element of GF(2^(2M)).

BRIEF DESCRIPTION OF THE DRAWINGS

[0022] The invention description below refers to the accompanying drawings, of which:

[0023]FIG. 1 is functional block diagram of a Galois Field multiplier constructed in accordance with the invention;

[0024]FIG. 2 is illustrates a functional block diagram of an alternative construction of the multiplier of FIG. 1;

[0025]FIG. 3 is a functional block diagram of a system for determining multiplicative inverses in GF(2^(2M)) that uses the multipliers of FIGS. 1 and 2;

[0026]FIGS. 4 and 5 are functional block diagrams of alternative systems for determining multiplicative inverses; and

[0027]FIG. 6 is a functional block diagram of a division circuit.

DETAILED DESCRIPTION OF AN ILLUSTRATIVE EMBODIMENT

[0028] To speed-up operations and reduce the complexity of hardware, we select a polynomial and a basis over GF(2^(2M)) such that a multiplication operation A*B=C, where the symbols are represented by their coefficients, for example, A=a₁₅, a₁₄, a₁₃ . . . , a₁, a₀, and B=b₁₅, b₁₄, b₁₃, . . . , b₁, b₀, involves producing for the coefficient c₀ a value that is a combination of the coefficients of A and B, and for the coefficients c₁, c₂ . . . c₁₅ the same combination of cyclically shifted versions of the coefficients of A and B. As discussed in more detail below with reference to FIG. 1, the multiplication operation is readily and quickly performed in hardware. First, however, we discuss the underlying theory by way of example over GF(2⁶).

[0029] A. The Selected Basis

[0030] In the example, the elements of GF(2¹⁶) are generated using the primitive polynomial:

g(x)=x ¹⁶ +x ¹⁵ +x ¹³ +x ¹² +x ¹¹ +x ¹⁰ +x ⁸ +x ⁷ +x ⁵ +x ³ +x ² +x+1

[0031] The field elements of GF(2¹⁶) are conventionally represented using the basis elements:

α⁰α¹α²α³α⁴α⁵α⁶α⁷α⁸α⁹α¹⁰α¹¹α¹²α¹³α¹⁴ and α¹⁵.

[0032] The element α⁵=(0000 0000 0010 0000) and the element α⁴⁰⁹⁸=(0100 0001 0000 1000) or the sum of:

(0100 0000 0000 0000)+(0000 0001 0000 0000)+(0000 0000 0000 1000)=α¹⁴+α⁸+α³

[0033] where rounded brackets are used herein to indicate field elements that are represented using the conventional basis.

[0034] The same field elements may be represented using a selected, or normal, basis of:

α¹α²α⁴α⁸α¹⁶α³²α⁶⁴α¹²⁸α²⁵⁶α⁵¹²α¹⁰²⁴α²⁰⁴⁸α⁴⁰⁹⁶α⁸¹⁹²α¹⁶³⁸⁴ and α³²⁷⁶⁸,

[0035] where each basis element is the square of the next lowest basis element. A given field element is represented using the selected basis as the combination of the selected basis elements that reproduce the element in the conventional basis. For example, the element α⁵=[0000 1001 1001 0000], where the square brackets are used herein to indicate that the element is represented using the selected basis. The element α⁵ is thus the sum of the basis elements α²⁰⁴⁸+α²⁵⁶+α¹²⁸+α¹⁶ or: $\begin{matrix} \left\lbrack 0000 \right. & 1000 & 0000 & {\left. 0000 \right\rbrack +} \end{matrix}\begin{matrix} \left\lbrack 0000 \right. & 0001 & 0000 & {\left. 0000 \right\rbrack +} \end{matrix}\begin{matrix} \left\lbrack 0000 \right. & 0000 & 1000 & {\left. 0000 \right\rbrack +} \end{matrix}\begin{matrix} \left\lbrack 0000 \right. & 0000 & 0001 & {\left. 0000 \right\rbrack = \begin{matrix} \left\lbrack 0000 \right. & 1001 & 1001 & \left. 0000 \right\rbrack \end{matrix}} \end{matrix}$

[0036] Using the conventional basis the sum of the field elements α²⁰⁴⁸, α²⁵⁶, α¹²⁸, α¹⁶ is: ${\begin{pmatrix} 1100 & 0101 & 1001 & 0100 \end{pmatrix} + \begin{pmatrix} 0001 & 1110 & 1110 & 0110 \end{pmatrix} + \begin{pmatrix} 0110 & 0110 & 1111 & 1101 \end{pmatrix} + \begin{pmatrix} 1011 & 1101 & 1010 & 1111 \end{pmatrix}} = \begin{pmatrix} 0000 & 0000 & 0010 & 0000 \end{pmatrix}$

[0037] which confirms that the representation of the element in the selected basis is correct. The representation of the element α⁴⁰⁹⁸=(0100 0001 0000 1000) in the selected basis can be determined from the conventional representation by taking the conventional basis elements that are included therein:

(0100 0000 0000 0000)+(0000 0001 0000 0000)+(0000 0000 0000 1000)

[0038] or

α¹⁴+α⁸+α³

[0039] and summing the representations of these elements in the selected basis:

α¹⁴=[0110 1001 1110 0000]

α⁸=[0000 0000 0000 1000]

α³=[0111 1000 0000 0000]

[0040] and the sum is [0001 0001 1110 1000]=α⁴⁰⁹⁸.

[0041] To then convert back to the conventional basis, determine the combination of elements that produce the symbol of interest in the selected basis:

α⁴⁰⁹⁸=α⁴⁰⁹⁶+α²⁵⁶+α¹²⁸α⁶⁴+α³²+α⁸

[0042] and using the conventional representations:

α⁴⁰⁹⁶=(0001 0000 0100 0010)

α²⁵⁶=(0001 1110 1110 0110)

α¹²⁸=(0110 0110 1111 1101)

α⁶⁴=(1011 1011 1000 0001)

α³²=(1001 0011 1101 0000)

α⁸=(0000 0001 0000 0000)

[0043] the sum is (0100 0001 0000 1000), or α⁴⁰⁹⁸.

[0044] As discussed in more detail below, using the selected basis simplifies certain Galois Field operations. For example, squaring an element represented using the selected basis is a cyclic rotation of the symbol by one bit. The element α⁶=[1111, 0000, 0000, 0000] is squared by cyclically rotating the symbol to the left:

(α⁶)²=[1110, 0000, 0000, 0001]=α¹².

[0045] Raising an element to a power 2^(k) is thus a cyclic rotation of the symbol by k bits to the left. Accordingly, raising an element to the power 256, or 2⁸, is a cyclic rotation to the left of eight bits.

[0046] B. Multiplication

[0047] Using the selected basis representations, multiplication of an element A by an element B, that is, A*B=C, is performed by first determining the coefficient c₀ as a sum of products of the coefficients of A and B:

c ₀ =a ₀*(b ₃ +b ₅ +b ₆)+a ₁*(b ₇ +b ₈)+a ₂*(b ₃ +b ₅+b₆ +b ₉ +b ₁₀ +b ₁₁ +b ₁₄ +b ₁₅) +a ₃*(b ₀ +b ₂ +b ₄ +b ₆ +b ₇ +b ₉ +b ₁₂ +b ₁₅)+a ₄*(b ₃ +b ₅ +b ₉ +b ₁₁) +a ₅*(b ₀ +b ₂ +b ₄ +b ₆ +b ₇ +b ₉ +b ₁₀ +b ₁₁) +a ₆*(b ₀ +b ₂ +b ₃ +b ₅ +b ₁₀ +b ₁₁ +b ₁₃ +b ₁₅) +a ₇*(b ₁ +b ₃ +b ₅ +b ₁₀ +b ₁₂ +b ₁₃)+a ₈*(b ₁ +b ₁₀) +a ₉*(b ₂ +b ₃ +b ₄ +b ₅ +b ₁₁ +b ₁₂ +b ₁₃ +b ₁₄)+a ₁₀*(b ₂ +b ₅ +b ₆ +b ₇ +b ₈ +b ₁₃) +a ₁₁*(b ₂ +b ₄ +b ₅ +b ₆ +b ₉ +b ₁₄)+a ₁₂*(b ₃ +b ₇ +b ₉ +b ₁₄) +a ₁₃*(b ₆ +b ₇ +b ₉ +b ₁₀)+a ₁₄*(b ₂ +b ₉ +b ₁₁ +b ₁₂) +a ₅*(b ₂ +b ₃ +b ₆ +b ₁₅)

[0048] The coefficient c₁ is determined by cyclically shifting the coefficients of each of A and B once to the left and combining them in the same manner:

c ₁ =a ₁*(b ₄ +b ₆ +b ₇)+a ₂*(b ₈ +b ₉)+a ₃*(b ₄ +b ₆ +b ₇ +b ₁₀ +b ₁₁ +b ₁₂ +b ₁₅ +b ₀) +a ₄*(b ₁ +b ₃ +b ₅ +b ₇ +b ₈ +b ₁₀ +b ₁₃ +b ₀)+a ₅*(b ₄ +b ₆ +b ₁₀ +b ₁₂) +a ₆*(b ₁ +b ₃ +b ₅ +b ₇ +b ₈ +b ₁₀ +b ₁₁ +b ₁₂) +a ₇*(b ₁ +b ₃ +b ₄ +b ₆ +b ₁₁ +b ₁₂ +b ₁₄ +b ₀) +a ₈*(b ₂ +b ₄ +b ₆ +b ₁₁ +b ₁₃ +b ₁₄)+a ₉*(b ₂ +b ₁₁) +a ₁₀*(b ₃ +b ₄ +b ₅ +b ₆ +b ₁₂ +b ₁₃ +b ₁₄ +b ₁₅)+a ₁₁*(b ₃ +b ₆ +b ₇ +b ₈ +b ₉ +b ₁₄) +a ₁₂*(b ₃ +b ₅ +b ₆ +b ₇ +b ₈ +b ₁₅)+a ₁₃*(b ₄ +b ₈ +b ₁₀ +b ₁₅) +a ₁₄*(b ₇ +b ₈ +b ₁₀ +b ₁₁)+a ₁₅*(b ₃ +b ₁₀ +b ₁₂ +b ₁₃) +a ₀*(b ₃ +b ₄ +b ₇ +b ₀)

[0049] The coefficient c_(j) is determined by cyclically shifting the coefficients of A and B by j bits, respectively, and combining them in the manner set forth above. The coefficients of A and B used to calculate c_(j) are thus a_(15+j mod16) . . . a_(0+j mod16), and . . . b_(15+j mod16) . . . b_(0+j mold16).

[0050] Referring now to FIG. 1, a Galois Field multiplier over GF(2¹⁶) includes cyclic shift registers 12 and 14 that are loaded with the coefficients of the selected basis representations of A and B, respectively. The shift register 12 thus holds a₁₅, a₁₄, . . . a₀, and shift register 14 holds b₁₅, b₁₄, . . . b₀. A computation circuit 16 combines the coefficients of A and B as set forth above to produce c₀. The shift registers are then shifted one bit to the left, such that they hold a₁₄, . . . a₀, a₁₅ and b₁₄, . . . b₀, b₁₅, respectively. Next, the combination circuit again combines the coefficients to produce c₁, and so forth.

[0051] Referring now to FIG. 2, to double the operating speed of the multiplier of FIG. 1, the computation circuit 12 is separated into two computation circuits 106 and 116. The combination circuit 106 produces the coefficients c₀, c₁ . . . c₇ at the same time that the circuit 116 produces the coefficients c₈, c₉ . . . c₁₅. Similarly, the combination circuit 16 may be further separated into four, eight or sixteen combination circuits that produce the corresponding coefficients in parallel, with associated reductions in the operating speed of the multiplier.

[0052] C. Subfields

[0053] Raising an element of GF(2¹⁶) to a power k*257 produces an element that is also an element of the subfield GF(2⁸). As an example, elements of GF(2¹⁶) that are also elements of the subfield GF(2⁸) are:

α²⁵⁷=[0,1,0,0, 0,0,0,0, 0,1,0,0, 0,0,0,0]=[<hex>4040]

α⁵¹⁴=[1,0,0,0, 0,0,0,0, 1,0,0,0, 0,0,0,0]=[<hex>8080]

α⁷⁷¹=[0,1,1,0, 1,1,0,1, 0,1,1,0, 1,1,0,1]=[<hex>6d6d]

[0054] Using the selected basis representations, the highest-order eight bits of a given symbol in GF(2⁸) are the same as the lowest-order eight bits of the symbol.

[0055] Raising an element of GF(2¹⁶) to a power k*4369 produces an element that is also an element of the subfield GF(2⁴). As an example, elements of GF(2¹⁶) that are also elements of the subfield GF(2⁴) are:

α⁴³⁶⁹=[0,0,0,1, 0,0,0,1, 0,0,0,1, 0,0,0,1]=[<hex>1111]

α⁸⁷³⁸=[0,0,1,0, 0,0,1,0, 0,0,1,0, 0,0,1,0]=[<hex>2222]

α¹³¹⁰⁷=[1,0,1,1, 1,0,1,1, 1,0,1,1, 1,0,1,1]=[<hex>BBBB]

[0056] Using the selected basis representations, the highest-order four bits of a given symbol in GF(2⁴), that is, bits with the subscripts 15, 14, 13 and 12, repeat four times as bits 11, 10, 9, 8; bits 7, 6, 5, 4; and bits 3, 2, 1 and 0.

[0057] Raising an element of GF(2¹⁶) to the power k*21845 produces elements of GF(2²), which are represented by symbols in which the two highest-order bits repeat eight times as bits 13, 12; bits 11, 10; . . . ; bits 1,0. For example,

α²¹⁸⁴⁵=[0,1,0,1, 0,1,0,1, 0,1,0,1, 0,1,0,1]=[<hex>5555]

α⁴³⁶⁹⁰=[1,0,1,0 1,0,1,0, 1,0,1,0, 1,0,1,0]=[<hex>AAAA]

[0058] D. Systems For Determining Multiplicative inverses

[0059] Using the selected basis and the elements of the various subfields, we have developed systems for determining the inverse of an element of GF(2^(2M)) that are efficient in terms of both the hardware and the size of an associated look-up table. The circuits are discussed in terms of examples in GF(2¹⁶).

[0060] Referring to FIG. 3, a circuit for determining the multiplicative inverse of an element A=α^(j) of GF(2¹⁶) includes an arithmetic element 302, such as a cyclic rotator, barrel shifter or multiplexer, that is hardwired to supply a symbol that is the equivalent of raising the element α^(j) to the power 256, or 2⁸, that is, cyclically rotating α^(j) to the left by 8 bits. The “rotation” is accomplished with essentially no delay. The register 302 thus supplies the element (a^(j))² ⁸ or a^(j*256), which is the conversion factor D discussed in U.S. Pat. No. 4,975,876 that is incorporated herein by reference, to a multiplier 304.

[0061] The multiplier 304 multiplies the element a^(j*256) by α^(j) to produce α^(j*257), which is also an element of the subfield GF(2⁸). As discussed above, the elements of GF(2⁸) that are represented using the selected basis have symbols in which the highest- and lowest-order eight bits are duplicates of one another. Accordingly, the multiplier 304 need produce only the eight highest- or eight lowest-order bits of the product α^(j)*α^(j*256). The multiplier may thus include only a portion of the computation circuitry 106 or 116 (FIG. 2).

[0062] The highest-order or the lowest-order eight bits produced by the multiplier 304 are used to enter a look-up table 306, which contains the highest or lowest-order eight bits of the multiplicative inverses of the elements of GF(2⁸). The table 306 thus contains 256 entries of eight bits each. The table produces the corresponding eight highest- or lowest order bits of α^(−j*257).

[0063] The eight bits produced by the table are supplied to the multiplier 308, which multiplies α^(−j*257) by α^(j*256) to produce the multiplicative inverse a^(−j). Accordingly, the multiplier loads the eight bits supplied by the table into the cyclic shift registers (FIG. 1 or 2) as the bits 15 to 8 and also the bits 7 to 0 of α^(−j*257). The multiplier 304 then operates as an 8-bit multiplier in the manner described above with reference to FIG. 1 or 2.

[0064] Referring now to FIG. 4, an alternative circuit for producing the multiplicative inverse of α^(j) includes the hardwired arithmetic element 302 that raises α^(j) to the power 256 to produce α^(j*256), the multiplier 304 that multiplies α^(j*256) by α^(j) to produce α^(j*257), or least the eight highest- and lowest order bits of α^(j*257). The system further includes an arithmetic element 402 that is hardwired to raise α^(j*257) to the power 2⁴. The register thus produces a symbol that is the equivalent of cyclically rotating α^(j*257) by 4 bits. The hardwiring thus interchanges bits 4 to 7 with bits 0 to 3, and the “rotation” is performed with essentially no delay.

[0065] The product produced by the arithmetic element 402, namely, α^(j*257*16) or α^(j*4112) is then multiplied by α^(j*257) in multiplier 404 to produce the value α^(j*4369), which is also an element of the subfield GF(2⁴). The multiplier 404 operates in the manner described above with reference to FIG. 2, to produce at least the four highest-order or lowest-order bits of α^(j*4369).

[0066] A 2⁴ or sixteen-entry look-up table 406 contains the four highest order bits of the multiplicative inverses of the elements of GF(2⁴). The table, which is entered using the four bits produced by the multiplier 402, produces the four highest- or lowest order bits of α^(−j*4369). In the example the table produces bits 15 to 12, and these four bits are supplied to a multiplier 408, which multiplies α^(−j*4369) by α^(j*4112) to produce α^(−j*257). The multiplier 408 thus uses the four bits supplied by the table as bits 15 to 12 and bits 11 to 8 of α^(−j*4369) and uses the eight highest order bits of α^(j*4112) that are supplied by the arithmetic element 402 as the contents of the shift registers 12 and 14 (FIGS. 1 and 2), and produces the eight highest order bits of α^(−j*257). The multiplier 308 then multiplies α^(−j*257) by α^(j*256) to produce α^(−j), as discussed above with reference to FIG. 3.

[0067] The system of FIG. 4 produces the multiplicative inverse of an element of GF(2¹⁶) using a look-up table 406 with only sixteen 4-bit entries. As discussed, the various multipliers, arithmetic elements and tables may instead produce and/or use the lowest order bits or other sets of repeating bits.

[0068] Referring now to FIG. 5, the size of the look-up tables 306 and 406 can be further reduced to 128 or 8 elements, respectively. An element P of the subfield GF(2^(k)), for example, k=8, has a multiplicative inverse P⁻¹=[p_(k−1), p_(k−2) . . . p₁, p₀]⁻¹ =[t_(k−1), t_(k−1) . . . t₁,t₀]. The element (P⁻¹)², is a cyclic shift or

(P ⁻¹)² =[[p _(k−1) , p _(k−2) . . . p ₁ p ₀]⁻¹]² =[p _(k−2) . . . p ₁ p ₀ p _(k−1),]⁻¹ =[t _(k−1) , t _(k−2) . . . t ₁ ,t ₀]² =[t _(k−2) . . . t ₁ ,t ₀ , t _(k−1)].

[0069] and thus, the multiplicative inverse of a given element P can be cyclically rotated to produce the multiplicative inverses of elements that are of the form P² ^(R) . Accordingly, a table of the multiplicative inverses of selected elements P can be used to produce the multiplicative inverses of the elements P² ^(R) .

[0070] We have selected for inclusion in the table 506 the multiplicative inverses of elements that have a 1 as a leading, or highest order, coefficient. The size of the table is thus reduced by one-half. Using elements of the subfield GF(2⁸), table has 128 elements. Before entering the table with an element P=A² ^(M) ⁺¹, the system, as necessary, cyclically rotates the coefficients of P in register 502 to produce an element with a leading coefficient of 1. The shift register 502 thus cyclically rotates the elements with leading coefficients of 0 “r” times to produce P² ^(R) which can then be used to enter the table 506. The table produces the multiplicative inverse (P² ^(R) )⁻¹, which a shift register 504 then rotates to the right r bits, that is, raises to the $\frac{1}{2^{R}}$

[0071] power, to produce ${\left\lbrack P^{{- 2}R} \right\rbrack^{\frac{1}{2^{R}}} = P^{- 1}},$

[0072] which in the example is the multiplicative inverse of A² ^(M) ⁺¹. The size of the look-up table is thus a trade-off with the circuitry and time needed to rotate the elements by r bits to the left to produce an element P² ^(R) with a highest-order coefficient of 1, and then to rotate P⁻² ^(R) by r bits to the right to produce P⁻¹.

[0073] Alternatively, the reduced-size table may contain the multiplicative inverses of the elements of the subfield GF(2^(k)) that have a 0 as the highest order coefficient, or those that have another bit set to a particular value.

[0074] The size of the look-up tables 306 and 406 may be even further reduced by including in the tables only the multiplicative inverses of elements P that have as the two highest order coefficients the pattern 1, 0. The table thus does not contain an entry associated with P=[0,0 . . . 0,0], which does not have a multiplicative inverse, or an entry associated with P=[1,1 . . . 1,1] which is its own multiplicative inverse. For all other elements, P=[. . . 10 . . . ] the system cyclically rotates the corresponding symbol by “s” bits in shift register 502, that is, raises the element to the power 2^(s), to produce P² ^(s) =[10 . . . ], where the two highest order coefficients are set to 1 and 0, respectively.

[0075] The system then enters the look-up table with the rotated symbol, to produce (P² ^(s) )⁻¹. The system next rotates (P² ^(s) )⁻¹ to the right by s bits in shift register 504, or raises (P² ^(s) )⁻¹ to the power

to produce P⁻¹. In the example using the subfield GF(2⁸), the table includes only 64 entries. Alternatively, the look-up table may include entries associated with elements of the subfield that have the two highest-order coefficients set to the pattern 0 and 1, respectively, or entries in which another set of two or more bits are set to a desired pattern. Referring now to FIG. 6, the systems of FIGS. 3-5 supply the element A⁻¹ to a multiplier 602. The multiplier then multiplies A⁻¹*B to produce $\frac{A}{B} = {K.}$

[0076] The multiplier 602 operates in the manner described above with reference to FIGS. 1 and 2.

[0077] The multiplication systems described above may be used with any Galois Field GF(2^(Q)) by representing the elements of the field using the selected, or normal, basis in which the basis elements are squares of one another. The systems for determining the multiplicative inverses may be used with any Galois Field GF(2^(2M)). 

What is claimed is:
 1. A method for producing a product C of two elements A and B of GF (2^(Q)), the method including the steps of: A. representing A and B using selected basis elements in which a given basis element is the square of the preceding basis element; B. combining the coefficients of A and B to produce a first set of coefficients of the product C; C. cyclically rotating the coefficients of A and B; D. combining the rotated coefficients of A and B to produce a next set of co-efficients of C; E. repeating steps C and D for the remaining sets of coefficients of C; and F. supplying the element c to encoding or decoding circuitry that encodes or decodes an associated codeword over GF(2^(Q)).
 2. The method of claim 1 wherein the step of representing includes converting a representation of A as a symbol using a conventional basis α⁰, α¹, α², α³ . . . to a representation as a symbol using the selected basis.
 3. The method of claim 1 further including the step of converting the representation of A as a symbol using the selected basis to a symbol using a conventional basis α⁰, α¹, α²,α³ . . .
 4. A method for producing an element of GF(2^(2M)) that is the multiplicative inverse of an element A of GF(2^(2M)), the method including the steps of: A. representing A using selected basis elements in which a given basis element is the square of the preceding basis element; B. cyclically rotating the coefficients of A by M bits to produce A² ^(M) ; C. multiplying A² ^(M) by A to produce A² ^(M) ⁺¹; D. entering a look-up table that contains the multiplicative inverses of elements of GF(2^(M)) and producing the multiplicative inverse of A² ^(M) ⁺¹; E. multiplying the multiplicative inverse of A² ^(M) ⁺¹ by A² ^(M) to produce the element of GF(2^(2M)) that is the multiplicative inverse of A; and F. supplying the multiplicative inverse to encoding or decoding circuitry that produces B/A as a step in encoding or decoding an associated codeword.
 5. The method of claim 4 wherein the step of entering the look-up table includes using the highest or lowest order M bits of A² ^(M) to enter the table.
 6. The method of claim 4 wherein the steps of multiplying each include the steps of a. combining the coefficients of the elements to be multiplied to produce a first coefficient of the product; b. cyclically rotating the coefficients of the elements to be multiplied; c. combining the rotated coefficients to produce a next coefficient of the product; and d. repeating steps b and c.
 7. The method of claim 4 wherein the steps of multiplying each include the steps of a. combining the coefficients of the elements to be multiplied to produce a first set of coefficients of the product; b. cyclically rotating the coefficients of the elements to be multiplied; c. combining the rotated coefficients to produce a next set of coefficients of the product; and d. repeating steps b and c.
 8. The method of claim 4 wherein the step of entering the look-up table further includes the steps of. a. cyclically rotating the coefficients of A² ^(M) ⁺¹ in a first direction to raise A² ^(M) ⁺¹ to a power 2^(s) to produce an element that has one or more particular coefficients set to a predetermined pattern, b. entering the table with the element produced in step a to retrieve the multiplicative inverse of the element, c. cyclically rotating the retrieved multiplicative inverse in a second direction to raise the multiplicative inverse to the power {fraction (1/25^(s))} to produce the multiplicative inverse of A² ^(M) ⁺¹.
 9. The method of claim 8 wherein the steps of multiplying each include the steps of a. combining the coefficients of the elements to be multiplied to produce a first coefficient of the product; b. cyclically rotating the coefficients of the elements to be multiplied; c. combining the rotated coefficients to produce a next coefficient of the product; and d. repeating steps b and c.
 10. The method of claim 8 wherein the steps of multiplying each include the steps of a. combining the coefficients of the elements to be multiplied to produce a first set of coefficients of the product; b. cyclically rotating the coefficients of the elements to be multiplied; c. combining the rotated coefficients to produce a next set of coefficients of the product; and d. repeating steps b and c.
 11. The method of claim 4 wherein the step of entering the look-up table fturther includes the steps of: a. cyclically rotating the coefficients of the element A² ^(M) ⁺¹ to raise the element to a power of 2, b. multiplying the result by A² ^(M) ⁺¹ to produce an element of a smaller Galois Field, c. entering a table that includes the multiplicative inverses of elements of the smaller Galois Field, d. multiplying the result of step c by the result of step a to produce the multiplicative inverse of A² ^(M) ⁺¹.
 12. The method of claim 11 wherein the steps of multiplying each include the steps of a. combining the coefficients of the elements to be multiplied to produce a first coefficient of the product; b. cyclically rotating the coefficients of the elements to be multiplied; c. combining the rotated coefficients to produce a next coefficient of the product; and d. repeating steps b and c.
 13. The method of claim 11 wherein the steps of multiplying each include the steps of a. combining the coefficients of the elements to be multiplied to produce a first set of coefficients of the product; b. cyclically rotating the coefficients of the elements to be multiplied; c. combining the rotated coefficients to produce a next set of coefficients of the product; and d. repeating steps b and c.
 14. The method of claim 4 wherein the step of representing includes converting a representation of A as a symbol using a conventional basis α⁰, α¹, α², α³ . . . to a representation as a symbol using the selected basis.
 15. The method of claim 4 further including the step of converting the representation of A as a symbol using the selected basis to a symbol using a conventional basis α⁰, α¹, α²,α³ . . . α² ^(M−1) .
 16. A system for multiplying two elements A and B of GF (2^(2M)) that are represented using a selected basis in which the basis elements are squares of one another, the system including: A. a first shift register for holding and cyclically rotating the coefficients of A; B. a second shift register for holding and cyclically rotating the coefficients of B; C. a computation circuit for producing one or more coefficients of the product C=A*B as one or more sums of terms that are combinations of the coefficients of A and B; the first and second shift registers providing the coefficients of A and B to the computation circuit for the computation of the coefficient c₀ and providing to the computation circuit for the computation of the coefficient c_(j) coefficients that have been cyclically shifted by j bits.
 17. The system of claim 16 wherein the computation circuit includes a plurality of computation sub-circuits that each produce an associated coefficient of the product, the plurality of sub-circuits producing in parallel a corresponding set of coefficients of the product.
 18. A system for producing an element of GF (2^(2M)) that is the multiplicative inverse of an element A of GF(2^(2M)) which is represented using a selected basis in which the basis elements are squares of one another, the system including: A. an arithmetic element hardwired to provide A² ^(M) as the element A cyclically shifted by M bits; B. a first multiplier for multiplying A² ^(M) by A to produce A² ^(M) ⁺¹; C. a look-up table that includes the multiplicative inverses of elements of GF(2^(M)), the system entering the look-up table with selected coefficients of A² ^(M) ⁺¹; and D. a second multiplier for multiplying the multiplicative inverse retrieved from the table by A² ^(M) to produce the element that is the multiplicative inverse of A.
 19. The system of claim 18 wherein one or both of the first and second multipliers includes a. a first shift register for holding and cyclically rotating the coefficients of a first element A² ^(M) ; b. a second shift register for holding and cyclically rotating the coefficients of a second element that is a power of A; c. a computation circuit for producing a coefficient of the product C as a sum of terms that are combinations of the coefficients of the first and second elements; the first and second shift registers providing the coefficients to the computation circuit for the computation of the coefficient c₀ of C and providing to the computation circuit for the computation of coefficients c_(j) coefficients of the first and second elements that have been cyclically shifted by j bits.
 20. The system of claim 19 wherein the computation circuit includes a plurality of computation sub-circuits that each produce an associated coefficient of the product, the plurality of sub-circuits producing in parallel a corresponding set of coefficients of the product.
 21. The system of claim 18 wherein the system further includes i. an arithmetic element for providing the element A² ^(M) ⁺¹ raised to a power 2^(M/2) as the element cyclically shifted by corresponding M/2 bits, ii. a third multiplier for multiplying the element (A² ^(M) ⁺¹)² ^(M/2) by A² ^(M) ⁺¹ to produce an element of a smaller Galois Field, iii. the look-up table contains the multiplicative inverses of elements of the smaller Galois Field and is entered using selected coefficients of the element of the smaller Galois Field, and iv. a multiplier for multiplying the multiplicative inverse produced by the table by the element A² ^(M) ⁺¹ raised to the power 2^(M/2).
 22. The system of claim 18 wherein the system further includes: d. a first shift register for cyclically rotating the coefficients of A² ^(M) ⁺¹ in a first direction to raise A² ^(M) ⁺¹ to a power 2^(s) and produce an element that has one or more particular coefficients set to a predetermined pattern, e. the look-up table contains the multiplicative inverses of the elements that have the particular coefficients set to the predetermined pattern, and the system enters the table using selected coefficients of A² ^(M) ⁺¹ raised to the power 2^(s), and f. a second shift register for cyclically rotating the coefficients of the element retrieved from the table to raise the element to the power {fraction (1/2^(s))} and produce the multiplicative inverse of A² ^(M) ⁺¹
 23. A system for producing an element of GF (2^(2M)) that is the quotient of B/A where A and B are elements of GF(2^(2M)) that are represented using a selected basis in which the basis elements are squares of one another, the system including: A. an arithmetic element hardwired to provide A² ^(M) as the element A cyclically shifted by M bits; B. a first multiplier for multiplying A² ^(M) by A to produce A² ^(M) ⁺¹; C. a look-up table that includes the multiplicative inverses of elements of GF(2^(M)), the system entering the look-up table with selected coefficients of A² ^(M) ⁺¹. D. a second multiplier for multiplying the multiplicative inverse retrieved from the table by A² ^(M) to produce the element that is the multiplicative inverse verse of A; and E. a third multiplier for multiplying B and the multiplicative inverse of A.
 24. The system of claim 23 wherein one or more of the multipliers include a. a first shift register for holding and cyclically rotating the coefficients of a first element; b. a second shift register for holding and cyclically rotating the coefficients of a second element; c. a computation circuit for producing a coefficient of the product C as a sum of terms that are combinations of the coefficients of the first and second elements; the first and second shift registers providing the coefficients to the computation circuit for the computation of the coefficient c₀ of C and providing to the computation circuit for the computation of coefficients c_(j) coefficients of the first and second elements that have been cyclically shifted by j bits.
 25. The system of claim 24 wherein the computation circuit includes a plurality of computation sub-circuits that each produce an associated coefficient of the product, the plurality of sub-circuits producing in parallel a corresponding set of coefficients of the product.
 26. The system of claim 23 wherein the system further includes i. an arithmetic element for providing the element A² ^(M) ⁺¹ raised to a power 2^(M/2) as the element cyclically shifted by corresponding M/2 bits, ii. a third multiplier for multiplying the element (A² ^(M) ⁺¹)² ^(M/2) by A² ^(M) ⁺¹ to produce an element of a smaller Galois Field, iii. the look-up table contains the multiplicative inverses of elements of the smaller Galois Field and is entered using selected coefficients of the element of the smaller Galois Field, and iv. a multiplier for multiplying the multiplicative inverse produced by the table by the element A² ^(M) ⁺¹ raised to the power 2^(M/2).
 27. The system of claim 26 wherein one or more of the multipliers include a. a first shift register for holding and cyclically rotating the coefficients of a first element; b. a second shift register for holding and cyclically rotating the coefficients of a second element; c. a computation circuit for producing a coefficient of the product C as a sum of terms that are combinations of the coefficients of the first and second elements; the first and second shift registers providing the coefficients to the computation circuit for the computation of the coefficient c₀ of C and providing to the computation circuit for the computation of coefficients c_(j) coefficients of the first and second elements that have been cyclically shifted by j bits.
 28. The system of claim 27 wherein the computation circuit includes a plurality of computation sub-circuits that each produce an associated coefficient of the product, the plurality of sub-circuits producing in parallel a corresponding set of coefficients of the product.
 29. The system of claim 23 wherein the system further includes: d. a first shift register for cyclically rotating the coefficients of A² ^(M) ⁺¹ in a first direction to raise A² ^(M) ⁺¹ to a power 2^(s) and produce an element that has particular coefficients set to a predetermined pattern, e. the look-up table contains the multiplicative inverses of the elements that have the particular coefficients set to the predetermined pattern, and the system enters the table using selected coefficients of A² ^(M) ⁺¹ raised to the power 2^(s), and f. a second shift register for cyclically rotating the coefficients of the element retrieved from the table to raise the element to the power {fraction (1/2^(s))} and produce the multiplicative inverse of A² ^(M) ⁺¹
 30. The system of claim 29 wherein one or more of the multipliers include a. a first shift register for holding and cyclically rotating the coefficients of a first element; b. a second shift register for holding and cyclically rotating the coefficients of a second element; c. a computation circuit for producing a coefficient of the product C as a sum of terms that are combinations of the coefficients of the first and second elements; the first and second shift registers providing the coefficients to the computation circuit for the computation of the coefficient c₀ of C and providing to the computation circuit for the computation of coefficients c_(j) coefficients of the first and second elements that have been cyclically shifted by j bits.
 31. The system of claim 30 wherein the computation circuit includes a plurality of computation sub-circuits that each produce an associated coefficient of the product, the plurality of sub-circuits producing in parallel a corresponding set of coefficients of the product. 